On a conjecture of Harris

Abstract: For $d \ge 4$, the Noether-Lefschetz locus $\mathrm{NL}_d$ parametrizes smooth, degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least $2$. A conjecture of Harris states that there are only finitely many irreducible components of the Noether-Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large $d$, but is true for $d \le 5$. She also showed that for $d=6, 7$, there are finitely many \emph{reduced}, irreducible components of $\mathrm{NL}_d$ of non-maximal codimension. In this article, we prove that for any $d \ge 6$, there are infinitely many \emph{non-reduced} irreducible components of $\mathrm{NL}_d$ of non-maximal codimension.